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Mirrors > Home > ILE Home > Th. List > opid | GIF version |
Description: The ordered pair 〈𝐴, 𝐴〉 in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.) |
Ref | Expression |
---|---|
opid.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
opid | ⊢ 〈𝐴, 𝐴〉 = {{𝐴}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 3458 | . . . 4 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | 1 | eqcomi 2092 | . . 3 ⊢ {𝐴, 𝐴} = {𝐴} |
3 | 2 | preq2i 3521 | . 2 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}} |
4 | opid.1 | . . 3 ⊢ 𝐴 ∈ V | |
5 | 4, 4 | dfop 3619 | . 2 ⊢ 〈𝐴, 𝐴〉 = {{𝐴}, {𝐴, 𝐴}} |
6 | dfsn2 3458 | . 2 ⊢ {{𝐴}} = {{𝐴}, {𝐴}} | |
7 | 3, 5, 6 | 3eqtr4i 2118 | 1 ⊢ 〈𝐴, 𝐴〉 = {{𝐴}} |
Colors of variables: wff set class |
Syntax hints: = wceq 1289 ∈ wcel 1438 Vcvv 2619 {csn 3444 {cpr 3445 〈cop 3447 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-un 3003 df-sn 3450 df-pr 3451 df-op 3453 |
This theorem is referenced by: dmsnsnsng 4903 funopg 5042 |
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